Mesh independence for nonlinear least squares problems with norm constraints. (English) Zbl 0771.65030

This paper considers application of the Gauss-Newton method to discretizations (based on Galerkin approximations) of infinite dimensional nonlinear least squares problems of the form \[ \min\| F(x)\|^ 2_ Y\quad\text{ under the constraint } \| x\|_ X\leq R. \] \(F\) is a sufficiently smooth weakly continuous function acting between the Hilbert spaces \(X\), the parameter space, and \(Y\), the output space. The analysis in the paper is local and assumes that a good estimate of the solution is available. The proposed variation applies the Gauss-Newton method to \(F(x)\) linearised round the current approximation \(x_ k\), but with the constraint unchanged.
A common observation is the stable behaviour for varying discretizations, and the constancy of the number of iterations to attain given terminating criteria. These phenomena are analysed and the mesh-independence for the constrained least squares problem is proved. In addition, sufficient conditions on the mesh independence are related to conditions which guarantee convergence of the Gauss-Newton method applied to the given problem. The paper concludes by illustrating the results with a numerical example, a two-point boundary value problem.


65K05 Numerical mathematical programming methods
65J15 Numerical solutions to equations with nonlinear operators
90C48 Programming in abstract spaces
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
90C30 Nonlinear programming
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